"I think that the Australian people are looking for answers and I hope that the political commentators will spell it out for them", she said.
She previously went up against Dutton and Scott Morrison for Liberal leadership on Friday, but was the first to be eliminated with just 11 votes out of the 85-member party room. As such, Morrison and his backers rallied their colleagues to vote for him over Bishop in the first round.
Renowned for her steely gaze dubbed the "death stare", Bishop's highlights as foreign minister include her strong condemnation of Russia's role in the shooting down of Malaysia Airlines MH17 in 2014.
His party will already have to contest a by-election for Mr Turnbull's harbour-side electorate in Sydney, traditionally a safe Liberal seat.
Taylor was instead given the energy portfolio in the new cabinet, taking over from now-treasurer Josh Frydenberg.
Bill Shorten is the preferred prime minister for the first time in three years, coming from trailing Malcolm Turnbull by 12 points to leading Mr Morrison 39 to 33.
Five things we learned in the Premier League
With Oxlade-Chamberlain out and Emre Can at Juventus, Keita looks like he could be just the midfielder that Klopp was looking for. The Egypt forward scored 44 times in his first season at Liverpool and took only 19 minutes to get off the mark at Anfield.
Mr Morrison became Australia's 30th prime minister on Friday in a leadership spill staged by top job-hopeful Peter Dutton.
MP Melissa Price, who represents major mining and agriculture centres in Western Australia state, picked up the environment ministry.
But Ms Ardern also acknowledged the work Malcolm Turnbull had done while he was in office.
Morrison, who was sworn Australia's Prime Minister on 24 August, chose to roll the cyber security functions into the Department of Home Affairs instead of appoint a replacement to Angus Taylor who resigned on 23 August.
"He's more of a known quantity", she told RN Breakfast.
Communications minister Mitch Fifield also made a return to the cabinet in the same position after resigning on Thursday, as did minister for human services Michael Keenan, who had his remit as assistant minister for digital transformation upgraded to a full portfolio.
Mr Turnbull is yet to announce his resignation from the seat but told reporters last week it would be "not before too long".